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Visually Representing the Landscape of Mathematical Structures

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Visually Representing the Landscape of Mathematical Structures Visually Representing the Landscape of Mathematical Structures Katherine Gravel Hayden Jananthan Jeremy Kepner Department of Mathematics Department of Mathematics Lincoln Laboratory Supercomputing Center Massachusetts Institute of Technology Vanderbilt University Massachusetts Institute of Technology Cambridge, Massachusetts Nashville, Tennessee Lexington, Massachusetts [email protected] [email protected] [email protected] Abstract—The information technology explosion has dramati- basic undergraduate courses to higher level subjects including cally increased the application of new mathematical ideas and has analysis, abstract algebra, and topology, visual intuition and led to an increasing use of mathematics across a wide range of tools for understanding still exist but become much sparser. fields that have been traditionally labeled “pure” or “theoretical”. There is a critical need for tools to make these concepts readily The lack of abundant, clear, visual explanations in abstract accessible to a broader community. This paper details the creation of visual representations of mathematical structures mathematics occurs naturally since abstract mathematics seeks commonly found in pure mathematics to enable both students to remove any dependency on the physical world and strives to and professionals to rapidly understand the relationships among make logical statements based only on generally stated rules and between mathematical structures. Ten broad mathematical [7]. Since the human brain is wired to process information structures were identified and used to make eleven maps, with visually, it is difficult to see how abstract math and visual 187 unique structures in total. The paper presents a method and implementation for categorizing mathematical structures and intuition may intersect [8], [9]. Therefore, there is a need to for drawing relationship between mathematical structures that create a representative visual that is both abstracted, to mirror provides insight for a wide audience. The most in depth map is the nature of pure mathematics, and specific, so individuals available online for public use [1]. may be able to associate objects represented in the map with Keywords—mathematics visualization, mathematics education, something tangible [10], [11]. pure mathematics Visual tools with the above mentioned ideas in mind have I. INTRODUCTION been created prior. For example, some individuals, generally It is often helpful to package information into visuals, which students, make small concept maps of different structures in are easily understood [2]. Visuals tools are used to enhance specific mathematical disciplines, or make concept maps of a understanding in media, education, professional circles, and group of related disciplines and exchange them on internet almost every other field. Many times, one may find it more forums [12], [13]. Although those maps may be useful to difficult to understand nonvisual media, especially when media the individuals who created them, they are of less utility to is attempting to convey complex or spatial information, such learning communities since they generally do not contain more as a direction manual for setting up a couch that does not than four words per node and do not explain the relationships come with pictures. Visuals representations are also of great between included concepts. There are also more rigorous use in education, specifically in mathematics, where they are representations such as a project by Wolfram Alpha, that able to ground abstract concepts in a more tangible medium: allows users to enter a mathematical structure into a search the physical world [3]–[6]. function that will return properties of the object, as well as a In mathematics as a whole, there are many visual intuitions local map of what it is related to [14]. Furthermore, in the book arXiv:1809.05930v1 [math.HO] 16 Sep 2018 and tools, although they are concentrated around less abstract Mathematics of Big Data there is extensive use of depictions mathematics. In grade school, one is taught to count on one’s of the abstract mathematics necessary to define an associative fingers, to ground the abstract concept of quantity in something array from more basic mathematical structures [15]. physical. In middle school and high school, graphs are relied upon heavily to aid in the understanding of slope as rise over Drawing inspiration from the graphics in Mathematics of run or representing integrals as the area under a curve. Even in Big Data, this project created a similar representations of undergraduate years, visual intuition still arises. For example, abstract structures while increasing the scope to a large group in linear algebra, matrix multiplication is represented by linear of fundamental structures in abstract mathematics. This project transformations. However, as mathematics students move from produced a catalog of different maps representing the connec- tions between abstract mathematical objects. It also categorizes This material is based in part upon work supported by the NSF under objects based on properties, gives a list of axioms sufficient grant number DMS-1312831. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do to constructing each structure and includes other features that not necessarily reflect the views of the National Science Foundation. make the map aesthetically appealing. II. BACKGROUND AND DEFINITIONS Types Informally, a mathematical obejct is any structure built from Previously defined objects. existing mathematical objects and structures, beginning with Functions sets as the basic building block. Matrices, functions, relations, Described functions (Definition 2.2) that must be and sets are all examples of mathematical objects. The terms defined over the structure. “object” and “structure” will be used interchangeably. For Relations clarity, we define what is meant by certain terms central to Enumerated the relations ((Definition 2.1) that must the project: be defined over the structure. Definition 2.1: Relation An n-ary relation R between sets Properties Detailed additional properties that functions and re- X1;:::;Xn is a subset of X1 Xn × · · · × lations over the structure must satisfy. Definition 2.2: Function An n-ary function f : X0 X Y is an (n + 1)-ary relation f X X× · · ·Y × For ease of presentation, the functions and relations on the n ! ⊆ 0 × · · · × n × where for each x0 X0; : : : ; xn Xn there exists a unique structures were not written in terms of set theoretic functions y Y such that (x 2; : : : ; x ; y) 2f. Such unique y is denoted (e.g. union, intersection) and relations (e.g. membership). 2 0 n 2 by f(x0; : : : ; xn) Although set theoretic definitions could be written in principle to ensure that the functions and relations over structures III. APPROACH really behaved as they claimed to, it would be cumbersome The first step was to survey pure mathematics as a whole, and hinder the definitions that are used more commonly in selecting objects that were central enough to the entire field literature and most fields of mathematics. that most pure mathematicians would know them, excluding Many structures included in the diagrams have multiple categories, which were too board to fall within the scope of the accepted and useful definitions. In most cases, it did not make project. The research involved a broad range of textbooks and sense to list more than one definition, since there was not a internet resources [15]–[19]. The next step was to determine consistent or elegant way to do so. In general, the definition a hierarchical relationship between all of the structures. The that would be presented earliest to a student learning in a hierarchical relationships were created by determining the traditional classroom. However, often the relationship between amount of structure encoded by the axioms of an object. For structures it was related to would be defined normally in terms example, there is more structure encoded in the axioms of a of an equivalent definition, so for consistency, the relationship module than those of a group. would need to be modified to be in terms of the definition The overall layout of the objects were facilitated with the stated within the node. The only exception depicting on the concept map program CMAP [20]. CMAP allows for rapid maps were Lattices, which have an algebraic and an order manipulation of visuals. Nodes were written to represent each theory defintion. Niether could be excluded because many object and labeled arrows were drawn between the nodes of the different types of lattices, both order theoretic and to represent their relationship to each other. Although all algebraic properties of the lattice were used in definitions. mathematical objects are related to all other mathematical While some structures have multiple accepted standard objects, drawing most connections would be redundant and definitions, there are other structures that have no standard clutter the diagram. Therefore, the connections that were definition. For example, some but not all authors require that drawin on maps were unique (with some exceptions disccused rings have a multiplicative identity. To deal with nonstandard later in the paper) Much of the thought of how the connections definitions, as few choices for definitions were made as should be drawn and what the placement of each structure possible. Although it would be possible to include multiple should encode was done at this stage. definitions, it was decided
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